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Unformatted text preview: PHYSICS 222
Introduction to Classical Physics II
Prof. Ruslan Prozorov
Iowa State University
Fall 2011 LECTURE 23
Power and resonance in RLC circuit power in a resistor (review)
v2 t 2
the instantaneous power is p t v t i t Ri t R
T energy dissipated in one cycle (over period T) is: U p t dt , T 0 with harmonic variation, 1 2 f i t I cos t Average power (over the cycle): 1 P p t dt I 2 R
T 0
2
T 1
2
I R
2
VRMS 2 cos 2 t dt 0 2 2
VRMS
I 2R
2 cos d 2 RI RMS R
0
2 V
I , I RMS 2
2 PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 2 exercise: LRC circuit – find R, L, C
VR RI R 3 Ohm
VC X C I X C 2 Ohm
VL X L I X L 6 Ohm 1
1
XC C 50 mF
C
X C
X L L L XL 0.6 mH 10 rad / s, I 10 A
4 PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 3 Power in an inductor PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 4 example: RL circuit
An RL circuit is driven by an AC generator
as shown in the figure. The current through
the resistor and the generator voltage are: A. always out of phase
B. always in phase
C. sometimes in phase and sometimes out
of phase E
VL And this is the current through all elements. PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University I
VR 17 October 2011 5 low and highpass filters
Vout Vout depends on frequency: C High smaller reactance VC = Vout 0 Low larger reactance no current flows
through R smaller VR VC = Vout Vout R This is a circuit that only passes
low frequencies: lowpass filter Bass knob on radio
0 1 RC If instead we look at the voltage through the resistor: highpass filter Treble control
(We can also use RL circuits are high and lowpass filters) PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 6 a  filter
1
XC C
X L L
if you want to filter any frequencies, but let alone the DC component:
• The capacitor C1 offers low reactance to the AC component of the input
while it offers infinite reactance to the DC component. As a result the
capacitor shunts an appreciable amount of the AC component while the DC
component continues its journey to the inductor L.
• The inductor L offers high reactance to the AC component but it offers almost
zero reactance to the DC component. As a result the DC component flows
through the inductor while the AC component is blocked.
• The capacitor C2 bypasses the AC component which the inductor had failed
to block. As a result only the DC component appears across the load RL. PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 7 the resonance
L Current amplitude in a series RLC circuit
driven by a source of amplitude E : I R C E
Z Maximum current when impedance Z XL XC R 2 is minimum
2 i.e., when XL XC L 0 1 C
1 LC Resonance:
Driving frequency = natural frequency PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 8 circuit behavior at resonance PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 9 tuning a radio PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 10 band pass filter
I E E cos Z R Maximum current E
R
I 0 maximum cos
(cos ~ 1) 0 0 1 2o ~ 0 (circuit in phase) LC Resonance
Low I ~ 0 due to capacitor
High I ~ 0 due to inductor PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 1 LC 17 October 2011 11 power in AC circuits
Instantaneous power supplied to the circuit: P t t i t E cos t I cos t Often more useful: Average power 1
P EI cos 2 P t EI cos t cos t cos t cos t cos t cos sin t sin cos t
cos2 x 1
2 cos x sin x Define: E rms E
2 Irms 2 0 1
2 cos2 xdx 2 0 1
2 cos x sin xdx 0 I
2 PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University P Erms Irms cos 17 October 2011 12 power factor
P Erms Irms cos Power factor (PF)
Maximum power = 0 I Resonant circuit E
E cos Z
R P Irms R
2 All energy dissipation happens
at the resistor(s). PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 13 the “quality” Q factor
How “sharp” is the resonance? (ie, the resonance peak) Umax
Q 2
U
• For RLC circuit, U max Umax is max energy stored in the system
ΔU is the energy dissipated in one cycle 1
LI 2
2 period 1 2
1 2 2
• Losses only come from R : U I RT I R 2
2 0 0 1 LC 0L XL,0
Q R
R
XC ,0
Q 0CR
R
1 PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University Q XL,0 XC ,0 R
R 17 October 2011 14 Q XL,0
R XC ,0
R Large Q sharp peak better “quality” L and C control how much energy is stored.
R controls how much energy is lost. Small resistance Large Q PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 15 transformers
Application of Faraday’s Law
• Changing EMF in primary coil creates changing flux
• Changing flux creates EMF in secondary coil.
V N Magnetic flux remains
mostly in the core.
Core “directs” B lines d B
for both coils
dt d B V1 V2 dt
N1 N2 V1 V2 N1 N2 V1 Efficient method to change
voltage for AC. N1 N2 V2 d B
dt If no energy is lost in the coils, power on both sides must be the same V1I1 V2I2 PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 16 transformers PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 17 Transformers PHYS222  Lecture 23  Prof. Ruslan Prozorov  Iowa State University 17 October 2011 18 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.
 Fall '09
 MEYER

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